Searching for the first near-Earth object family

Eva Schunová a,!, Mikael Granvik b,e, Robert Jedicke b, Giovanni Gronchi c, Richard Wainscoat b,Shinsuke Abe d

a Department of Astronomy, Physics of the Earth and Meteorology, Comenius University, Mlynská dolina, Bratislava 942 48, Slovakiab Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, United Statesc Dipartimento di Matematica, University of Pisa, Piazzale B. Pontecorvo 5, 56127 Pisa, Italyd Institute of Astronomy, National Central University, No. 300, Jhongda Rd., Jhongli, Taoyuan 32001, Taiwane Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland

a r t i c l e i n f o

Article history:Received 9 March 2012Revised 27 June 2012Accepted 27 June 2012Available online 8 July 2012

Keywords:Near-Earth objectsAsteroidsDynamics

a b s t r a c t

We report on our search for genetically related asteroids amongst the near-Earth object (NEO)population—families of NEOs akin to the well known main belt asteroid families. We used the techniqueproposed by Fu et al. (Fu, H., Jedicke, R., Durda, D.D., Fevig, R. Binzel, R.P. [2005]. Icarus 178(2), 434–449)supplemented with a detailed analysis of the statistical significance of the detected clusters. Their signif-icance was assessed by comparison to identical searches performed on 1000 ‘fuzzy-real’ NEO orbit distri-bution models that we developed for this purpose. The family-free ‘fuzzy-real’ NEO models maintain boththe micro and macro distribution of five orbital elements (ignoring the mean anomaly). Three clusterswere identified that contain four or more NEOs but none of them are statistically significant at P 3r.The most statistically significant cluster at the ! 2r level contains four objects with H < 20 and all mem-bers have long observational arcs and concomitant good orbital elements. Despite the low statistical sig-nificance we performed several other tests on the cluster to determine if it is likely a genetic family. Thetests included examining the cluster’s taxonomy, size–frequency distribution, consistency with a family-forming event during tidal disruption in a close approach to Mars, and whether it is detectable in a properelement cluster search. None of these tests exclude the possibility that the cluster is a family but neitherdo they confirm the hypothesis. We conclude that we have not identified any NEO families.

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1. Introduction

We report here on our null search for a statistically significantcluster of genetically related near-Earth objects (NEO). The identi-fication of such a NEO ‘family’ would enable further research intothe physical characteristics of NEOs as was the case with the iden-tification of asteroid families in the main belt, Jupiter Trojan andtrans-neptunian minor planet populations. The physical character-istics of the ensemble of NEO family members—taxonomic andmineralogical types, sizes, rotation periods, shapes, pole orienta-tions, existence of satellites, etc.—would lead to a better under-standing of their morphology and the mechanisms affecting theirdynamical and collisional evolution. If the family members havea small minimum orbital intersection distance (MOID) with theEarth (or any other planet) their presence on a sub-set of danger-ous orbits will increase the total impact probability over what iscurrently understood because the enhancement in orbit elementphase space is not incorporated into contemporary NEO models.

In one sense there are already some known NEO families sinceassociations have been proposed between several meteor showersand their presumed parent body NEOs (both asteroids and comets,e.g. Sekanina, 1973; Porubcan et al., 2004). The most known andwell-accepted is the connection between the unusual B-type NEO(3200) Phaethon and the Geminid meteor complex Ohtsuka et al.(2008). However, the associations between meteors and NEOs arenot families in the traditional understanding. The known asteroidfamilies are produced through the catastrophic disruption of a par-ent asteroid in a severe impact with another asteroid. The discov-ery of genetically related pairs of asteroids in the main belt (e.g.Vokrouhlicky and Nesvorny, 2008) suggests that other mecha-nisms may produce asteroid families such as the spin-up and rota-tional fission of rapidly rotating objects or the splitting of unstablebinary asteroids. Meteoroids are the small-end tail of those familycreation mechanisms but can also be produced through surfaceejection driven by the volatilization of sub-surface ices on cometsor ‘active asteroids’. While no formal distinction exists between themeteor ‘families’ and the asteroid families the size ratio betweenthe largest and second largest fragments nicely divides the samplesand creation mechanisms.

0019-1035/$ - see front matter Published by Elsevier Inc.http://dx.doi.org/10.1016/j.icarus.2012.06.042

! Corresponding author. Fax: +1 808 988 2790.E-mail address: schunova@ifa.hawaii.edu (E. Schunová).

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No NEO families have been identified and confirmed. Drum-mond (2000) suggested that there were many ‘associations’ inthe NEO population but Fu et al. (2005) showed that all of themwere likely chance alignments of their orbits. The dearth of NEOfamilies is in contrast to the more than 50 families known in themain belt (e.g. Nesvorny et al., 2005). Indeed, Hirayama (1918) pro-posed the existence of main belt asteroid families when there wereonly 790 known main belt asteroids while there are now morethan 7500 known NEOs.

Asteroid families are typically identified by the similarity of thefamily member’s orbital elements. Several metrics have beendeveloped to quantify the orbital similarity (e.g. Southworth andHawkins, 1963; Valsecchi et al., 1999; Jopek et al., 2008; Vokrouh-licky and Nesvorny, 2008). The best known is the Southworth–Hawkins D-criterion (DSH; Southworth and Hawkins, 1963) thatwas developed and used to search for parent comets of meteorstreams and to link meteor streams with NEOs (e.g. Gajdo! andPorubcan, 2005; Ohtsuka et al., 2008).

It is important to keep in mind that simply identifying memberswithin a population with similar orbital elements does not meanthat they are actually a genetically related family, i.e. that theydo not necessarily derive from a single parent object. As the num-ber of objects in the population increases there is a correspondingincrease in the likelihood that chance associations will mimic fam-ilies and extra care must be taken to establish a proposed family’sstatistical significance.

The major difference between the ability to identify families inthe NEO and main belt populations is that the orbits of asteroids inthe main belt are stable on timescales comparable to the age ofSolar System. Their long term stability allows the calculation ofmost objects’ proper elements (e.g. Knezevic et al., 2002), essen-tially time-averaged orbital elements, that are better suited tothe identification of similar orbits that are the hallmarks of aster-oid families. Furthermore, the non-gravitational evolution of themain belt asteroids’ orbits is slow under the influence of the Yar-kovsky effect (e.g. Vokrouhlicky et al., 2006a) so that they occupythe same orbital element phase space for long periods of time.

The main belt families identified by the similarity in their orbitelements range in age from millions to billions of years. Spectro-scopic campaigns have confirmed that the members of older fam-ilies typically share the same spectral type; as expected if they allderive from the same parent body and/or are covered by similarregolith during the family formation event (e.g. Cellino et al.,2002; Ivezic et al., 2002a). Near Earth objects on the other hand re-side in a turbulent dynamical environment with average lifetimesof ! 106 year under the strong gravitational influence of the terres-trial planets and Jupiter (e.g. Morbidelli et al., 2002; Gladman et al.,1997). Thus, the calculation of NEO proper elements is difficult be-cause they may cross the orbits of the planets during their timeevolution (Gronchi and Milani, 2001). The resulting NEO properelements are only valid for the time between their close encoun-ters with the planets and when mean motion resonances of low or-der with the planets do not occur.

The search for similar orbits amongst the NEO population usingonly the osculating elements is even more limited in the timeframe over which the orbits maintain their coherence (Pauls andGladman, 2005). Furthermore, the non-gravitational accelerationsacting on NEOs are generally much stronger than those acting onthe more distant asteroids of the Solar System. Thus, if a NEO fam-ily exists it will only be possible to identify it for a short time per-iod after its creation as its members rapidly disperse.

NEO families might form by several imaginable methods (Fuet al., 2005):

" spontaneous disruption (e.g. YORP spin-up, internal thermalstresses)

" tidal disruption during close approach to a planet" intra-NEO collisions" collision with a smaller main belt asteroid

It is difficult or impossible to assign a likelihood to the differentformation mechanisms. While 11% of comets spontaneously dis-rupt (e.g. Massi and Foglia, 2009) as they enter the inner Solar Sys-tem #1:0 AU < q 6 1:5 AU$, presumably due to thermal stressesinduced by approaching the Sun, the NEOs have typically residedin the terrestrial planet zone for long periods of time and manyorbital periods. They are thus unlikely to disrupt by this mecha-nism. On the other hand, simulations suggest that the YORP ther-mal torque can increase an asteroid’s spin rate to the pointwhere it spontaneously sheds material and some of this materialmay be ejected faster than the parent body’s escape speed (e.g.Walsh and Richardson, 2006; Pravec et al., 2010; Jacobson andScheeres, 2011). If the process is repeated multiple times over ashort time span it might be possible to create an asteroid familyin this manner. Families created in this way might be distinguishedby their rotation rates, mass ratios, or pole orientations of theirmembers.

Tidal disruption of NEOs may occur when they pass close to aplanet (e.g. Richardson et al., 1998; Walsh and Richardson, 2006;Walsh and Richardson, 2008) as occurred in the production ofthe family of objects associated with Comet Shoemaker–Levy-9(Sekanina et al., 1994). While the tidal disruption of asteroids hasbeen simulated under a variety of conditions (e.g., spin rate, poleorientation, closest approach distance, encounter speed), there isno estimate of how many tidal disruptions actually occur.

The likelihood of forming NEO families with the third mecha-nism is probably relatively small, since the space number-densityof NEOs is very small compared to the space number-density ofmain belt asteroids (Bottke et al., 1996).

Perhaps the most likely formation mechanism for a NEO familyis a collision between the NEO and a smaller main belt asteroid(Bottke et al., 1996). Many NEOs remain on orbits with aphelia inor beyond the main belt where they are typically slower than otherasteroids at the same distance and therefore suffer higher speedcollisions compared to those between two main belt asteroids atthe same heliocentric distance. Thus, smaller projectiles could beeffective impactors at catastrophically disrupting a NEO.

In summary, NEO families will provide to the opportunity ofexploring the physics of asteroid disruptions at different scalesand for different reasons than those observed in the main belt.

2. Method

We searched for families in the NEO population using the meth-od proposed by Fu et al. (2005). The technique identifies subsets ofobjects with similar orbits within a population. We used theirosculating elements, because of the problems involved in calculat-ing NEO proper elements. We are thus limited to identifying NEOfamilies that have formed relatively recently though this is notreally a problem because (1) NEO dynamical lifetimes1 are on theorder of a million years and (2) young families are interesting.

We used the Southworth–Hawkins DSH criteria (Section 2.1) toquantify the similarity between two orbits. It could be argued thatother criteria would yield better results but this criteria has beenwell tested and forms the basis of the Fu et al. (2005) technique.A set of NEOs that all have mutual DSH criteria below a thresholdDcluster is considered a ‘cluster’. The technique allows for and favorssub-clustering within the cluster under the assumption that therecould be a tight ‘core’ within a cluster surrounded by a looser

1 A NEO’s dynamical lifetime is the time during which it remains in the NEO region.

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1051

assemblage of related objects. We used the Minor Planet Center(MPC) mpcorb.dat orbit element data set that contained 7563NEOs as of March 2011 (through 2011 DW). The technique isdescribed in detail below.

The real difficulty in identifying NEO families is not the identi-fication of the clusters but in establishing their statistical signifi-cance. Indeed, as shown by Fu et al. (2005) for NEOs and Paulsand Gladman (2005) for fireballs, it is surprisingly difficult to provethat similar orbits are statistically significant within a population.We first tested our method on synthetic family-free NEO orbitmodels but then developed more realistic family-free NEO modelsderived from the known NEO population. We used multiple in-stances of the family-free NEO models to establish the statisticalsignificance of our NEO clusters.

2.1. The Southworth–Hawkins D-criterion

The Southworth–Hawkins DSH metric quantifies the similarityof two orbits using five orbital elements (Southworth and Hawkins,1963). We use the DSH metric in its original form because it has anestablished pedigree (e.g. searching for parent comets of meteorstreams) that has been used to successfully identify members ofknown meteor streams (Southworth and Hawkins, 1963; Sekanina,1970) and main belt asteroid families (Lindblad and Southworth,1971).

The DSH metric between two orbits denoted with subscripts mand n is defined by:

DSH %!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d2

1 & d22 & d2

3 & d24

q; #1$

with

d1 %qm ' qn

AU; #2$

d2 % em ' en; #3$

d3 % 2 sin#I=2$; #4$

d4 % #em & en$ sin#P=2$; #5$

I % arccos cos im cos in & sin im sin in cos#Xm 'Xn$( ); #6$

P % #xm 'xn$ * 2 arcsin cosim & in

2sin

Xm 'Xn

2sec

I2

" #; #7$

where q % a#1' e$ is the perihelion distance, a is the semi-majoraxis, e is the eccentricity, i the inclination, X is the longitude ofthe ascending node and x is the argument of perihelion. I repre-sents the angle between the poles of the two orbits and P repre-sents the angle between their perihelia directions (Drummond,2000). Use the positive sign for the arcsin term in Eq. (7) whenjXm 'Xnj 6 180+ and the negative sign otherwise.

2.2. Identifying clusters in orbital element space

We adopted the cluster identification method developed by Fuet al. (2005) that in turn incorporated the techniques describedby Drummond (2000). A ‘cluster’ is a grouping of objects withmutually similar orbits whereas we will reserve the term ‘family’for a cluster with high statistical significance that contains mem-bers that are likely genetically related. Fu et al. (2005) showed inlimited testing that the method is capable of identifying syntheticNEO families with minimal contamination.

We grouped objects into clusters based on the values of fourparameters that are described in greater detail below:

" Dcluster: the maximum DSH between any pair of objects in acluster." Dpair: the maximum DSH between ‘tight’ pairs in a cluster." SCRmin: the minimum String length to Cluster size Ratio." PFmin: The minimum fraction of pairs of asteroids in the cluster

with DSH < Dpair .

We identified candidate clusters as sets of N objects with mutualDSH < Dcluster . Within each candidate cluster we then identified allpairs (n) of asteroids with DSH < Dpair . The pair fraction PF is thenumber of detected pairs divided by the number of all possible pairs

in the cluster PF % n N#N'1$2

.$ %and we required that all our clusters

satisfy PF > PFmin. Then, within each candidate cluster we deter-mined the maximum string length L—the maximum number of ob-jects that are connected in a continuous pair-wise fashion such thateach sequential pair in the string satisfies DSH < Dpair . The SCR is theratio of the number of objects in the string to the number of objectsin the cluster, SCR % L=N, and our final set of clusters all satisfySCR > SCRmin. Note that a string must contain more than two objects.

The goal of our selection criteria is to identify tight clusters ofobjects in orbital element space but the PF and SCR cuts recognizethat as NEOs undergo rapid dynamical and non-gravitational evo-lution some of the members may evolve quickly onto different or-bits. The pair and string searches allow for a tight ‘core’ of objectswith a periphery of other objects.

2.3. Selecting thresholds for the cluster identification algorithm

While it is simple to verify that our algorithm can identify ob-jects in clusters with similar orbital elements it is not trivial to se-lect the four threshold parameters (Dcluster , Dpair , SCRmin; PFmin) tomaximize the cluster detection efficiency while minimizing thecontamination by false positives. This required a family-free NEOorbit distribution model that also incorporated the observationalselection effects typical of the asteroid surveys contributing thecurrent NEO inventory. The selection effects are important becausee.g. they favor the discovery of objects on Earth-like orbits andtherefore increase the orbital element phase-space density ofknown objects on these types of orbits.

2.3.1. Synthetic family-free NEO modelTo generate our synthetic family-free NEO model we started

with the set of NEOs from the Synthetic Solar System Model(S3M; Grav et al., 2011). The S3M includes over 11 million objectsranging from those that orbit the Sun entirely interior to theEarth’s orbit to the most distant reaches of the Solar System. The268,896 NEOs with absolute magnitude H < 25:0 in the S3M weregenerated in accordance with the #a; e; i$ orbital element residence-time distribution of the Bottke et al. (2002) NEO model. The modeldoes not include any NEO families.

To model the observational selection effects on the S3M NEOpopulation we performed a long-term survey simulation usingthe Pan-STARRS Moving Object Processing System (Denneauet al., 2007). MOPS was developed to process source detection datafrom Pan-STARRS (Kaiser et al., 2010) but also incorporates real-time processing of synthetic detections to monitor the system’sperformance. Thus, it can be also used as a pure survey simulator.

In an ecliptic longitude (k0) and latitude (b) system centered onthe opposition point #k0; b$ % #0+;0+$ the MOPS simulated survey2

2 The simulated survey discussed herein is only loosely related to the actual PS1survey. The details of the survey simulation are notimportant—all that matters is thatthe simulated survey reproduces the observational selection effects of the ensembleof surveys that produced the known NEO population. The litmus test is whether theresulting simulated orbit element distributions match the known orbit distributions.

1052 E. Schunová et al. / Icarus 220 (2012) 1050–1063

region is broken into two regions covering about 5500 deg2: (1) theopposition region with jk0j < 30+ and jbj < 40+ and (2) two ‘sweetspots’ with jbj < 10+ and 60+ < jk0j < 90+.

MOPS uses a full N-body ephemeris determination to calculatethe exact (RA;Dec) of every NEO in each synthetic field and thendegrades the astrometry to the realized PS1 astrometric error levelof 0:100 (Milani et al. (2012) have shown that PS1 achieves ! 0:1300

absolute astrometric error). The photometry for each object isdegraded in a S/N-dependent manner such that as S/N ? 5 themagnitude error approaches !0.1 mag. MOPS then makes a cutat S/N = 5 to simulate the statistical loss of detections near thePS1 system’s limiting magnitude1 of R ! 22:7. Each field is ob-served twice each night within !15 min to allow the formationof tracklets, pairs of detections at nearly the same spatial location,that might represent the same Solar System object. Fields arere-observed three times per lunation (simulated weather permit-ting) and tracklets are linked across nights to form tracks thatare then tested for consistency using an initial orbit determination(IOD).

Detections in tracks with small astrometric residuals in the IODare subsequently differentially corrected to obtain a final orbit. Our4-year MOPS survey simulation on the S3M NEO populationyielded 8020 derived objects (synthetic discoveries). Fig. 1 showsthat the derived synthetic NEOs are a rough match to the knownNEOs. Thus, our NEO survey simulation yields a set of syntheticNEOs that are a proxy for the known population including theirobservational selection effects (Jedicke et al., 2002). Pan-STARRSis currently operating a single prototype telescope on Haleakala,Hawaii, known as PS1.

2.3.2. Cluster identification in the synthetic NEO modelWith a large amount of computing time we could identify the

set of clusters detectable in the real NEO population for all possiblecombinations of the four thresholds—Dcluster , Dpair , SCRmin and PRmin.But this is impractical because we will also need to run orders ofmagnitude more tests to establish the detected clusters’ statisticalsignificance. We were guided in our selection of the thresholds byprevious work, our own experience, and testing the algorithm onthe synthetic NEO model.

Fig. 2 shows the results of the cluster identification method ap-plied to the family-free synthetic NEO model using Dcluster % 0:060and Dpair % 0:058 without any cuts on SCR and PF. Even though ourDSH thresholds are already considerably tighter than those pro-posed by Fu et al. (2005) we still identify many false clusters con-taining three members. We even identify one false 5-membercluster with SCR % 0:8 and PF % 0:3! However, the number of falseclusters drops quickly with the number of cluster members (42triplets, seven 4-member clusters and only one 5-member cluster).As the synthetic model is a good representation of the real NEOpopulation we expect roughly the same number of false clustersamongst the real NEOs when we use the same Dcluster—Dpair cuts.Thus, even though we will explore the full range of tighter thresh-olds on the Dcluster and Dpair values as described below, based onFig. 2 we will use SCRmin % 0:75 and PFmin % 0:5 to identify clusterscontaining P 4 members and not inspect the clusters containingNEO pairs and triplets. Establishing the smaller clusters’ statisticalsignificance will be difficult or impossible using the techniquesdeveloped here.

In the following Section 2.4 we will use the thresholds derivedhere to identify real NEO families for detailed analysis. However,when we measure the statistical significance of those NEO familiesin Section 3 we will abandon the synthetic model in favor of a morerealistic one, but the more realistic model must be created usingthe thresholds derived here.

2.4. NEO clusters in the real population

We identified three clusters of four or more members in the realNEO population from the mpcorb.dat database.3 (We will ignorethe 13 triplets and 243 pairs identified with the same cuts.) Themembers’ orbital elements and other physical parameters are pro-vided in Tables 1 and 2. The three clusters are labelled C1, C2 andC3, and have 4, 6 and 5 members respectively. The absolute magni-tudes of the cluster members spans 18:5 < H < 29:5 correspondingto diameters ranging from several meters up to several hundredsof meters depending on the choice of albedo.

The total number of P4 member clusters identified in the realdata should be compared to the 0 P 4 member clusters identifiedin the family-free synthetic data with SCRmin % 0:75 and PFmin % 0:5(see Fig. 2). The disparity in the number of P4-member clusterscould be due to the presence of real NEO families but we will arguebelow that it is due to the lack of fidelity in the synthetic NEOmodel. We find that the synthetic NEO model must be exquisitelymatched to the real NEO population in order to assess the statisti-cal significance of the detected clusters.

The C1 cluster is the only one composed of objects with H < 20and, more importantly, the only one containing objects with orbitalarcs longer than 100 days. The C1 cluster members’ orbital elementuncertainties are typically 1–2 orders of magnitude lower than theother two clusters. All the C1 members belong to the Amor4 NEOsub-population so that they do not cross the Earth’s orbit. Indeed,the members of this cluster have a perihelion distance of !1.25 AUimplying that if the cluster is a genetic NEO family it has probablynever approached close to the Earth or Venus. Considering the sim-ilarity of all five orbital elements the cluster must have formed rel-atively recently and it is unlikely that the cluster or its memberscould have approached the Earth and then evolved onto orbits thatdo not cross the Earth’s on a short time scale.

In contrast to the C1 cluster, the C2 and C3 clusters are com-posed of small objects with H > 21:1 and H > 27:7 respectivelyand include objects with orbital arcs sometimes spanning just sev-eral days. The short arc lengths yield large uncertainties on theorbital elements which in turn induce a large uncertainty in theC2 and C3 DSH. The clusters thus illustrate how false associationscan arise because of the orbit element uncertainties. The nominalorbits of the two clusters place them in the Apollo NEO sub-population with the C2 cluster lying close to the Amor–Apollotransition and the C3 cluster close to the Apollo–Aten transition.Their location near the transition regions is not a coincidence,these small objects were identified by NEO surveys only becausetheir orbits bring them very close to Earth. The perihelion distanceis !1.00 AU for the members of the C2 cluster while the membersof the C3 cluster are on very Earth-like orbits with a ! 1:00 ande < 0:1. Establishing the C2 and C3 clusters’ statistical significancewould be difficult because observational selection effects are notwell-characterized for objects in their size range and this inducesa large uncertainty in the orbit and size-distribution models forsmall NEOs.

Given the problems with the C2 and C3 clusters we will concen-trate on establishing the statistical significance of the C1 cluster.One method of doing so is to run the cluster finding algorithmon many instances of the synthetic family-free NEO model de-scribed in Section 2.3.1. If we identify 63 false clusters of P4members in 1000 realizations of the synthetic NEO model we couldclaim that C1 is statistically significant with P99.7% or P3r con-fidence. We did not employ this technique because of our concerns

3 http://www.minorplanetcenter.net/iau/MPCORB.html.4 The Amors have perihelion distance q in the range 1:0167 AU < q 6 1:3 AU,

Apollos have a > 1:0 AU and q 6 1:0167 AU and the Atens have a < 1:0 AU andaphelion Q > 0:983 AU.

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1053

Fig. 1. Semi-major axis, eccentricity and inclination distributions of the real NEO population (solid line), the synthetic NEO model (dotted line) and our fuzzy-real NEO model(data points).

1054 E. Schunová et al. / Icarus 220 (2012) 1050–1063

with the use of the !10 year old Bottke et al. (2002) NEO modelthat underestimates the number of Amor-type NEOs like the mem-bers of the C1 cluster. There are currently 474 known Amors withH < 18 (as of March 2011) compared to the Bottke et al. (2002)prediction of 310* 38—a >4r difference between the real and syn-thetic NEO populations. If the Bottke et al. (2002) NEO modelunderestimates the number of Amor-type NEOs then it would im-ply that we will overestimate the C1 cluster’s statistical signifi-cance. Furthermore, the synthetic NEO model relies on a surveysimulation that was not intended to perfectly model real surveysand yields !6% more objects than the real NEO population withsmall but perhaps significant skewing in the synthetic orbital

element distributions as shown in Fig. 1. We need a better syn-thetic NEO model as described in the next section.

2.4.1. Fuzzy real NEO modelsThe main problem with the Bottke et al. (2002) synthetic NEO

model is illustrated in Fig. 3—there is a huge discrepancy betweenthe normalized DSH distributions for the closest pairs within thereal and synthetic NEO populations. But establishing the statisticalsignificance of our NEO clusters requires a large number of inde-pendent high-fidelity family-free NEO models that incorporateobservational selection effects. Thus, we developed NEO modelsusing a technique that (i) maintains both the micro and macro

Fig. 2. Pair fraction (PF) vs. String to Cluster size Ratio (SCR) for false clusters detected in the family-free synthetic NEO model when Dcluster % 0:060 and Dpair % 0:058. The areaof the filled circles is proportional to the number of false clusters at each PF–SCR combination. Black circles represent clusters with 3 members, red is for clusters with 4members, and blue represents clusters with P5 members. Empty black, red and blue circles represent the quantized possible values for 3, 4 and P5 member clustersrespectively. It is impossible to have the combinations of PF and SCR on the left side of the figure. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Table 1Orbital elements for the members of 3 NEO clusters (C1, C2 and C3) with four or more members. Members of the clusters are listed in order of decreasing size. None of the clustersare statistically significant at > 3r. Data is from the JPL Small-Body Database Browser http://ssd.jpl.nasa.gov/sbdb.cgi.

Name a (AU) Da #,10'7$ e De #,10'6$ i (!) Di #,10'5$ x (!) Dx #,10'4$ X (!) DX #,10'4$

C12000 HW23 2.154 1 0.424 1 7.76 7 245 2 47 12006 JU41 2.123 5 0.429 2 7.61 8 236 3 52 22001 PF14 2.120 4 0.410 0.5 6.78 2 254 3 38 12008 LN16 2.141 5 0.421 0.2 7.82 2 261 4 36 2

C21999 YD 2.463 4000 0.593 70 1.38 12 62 0.2 10 202003 UW5 2.470 2, 105 0.577 300 1.87 77 52 0.2 17 102008 UT5 2.279 2, 105 0.555 400 1.61 91 33 0.1 38 102008 YW32 2.319 5, 105 0.559 104 0.99 125 2 0.6 71 402007 YM 2.584 2,107 0.617 3,105 0.99 2,105 11 0.8 60 7,104

2005 WM3 2.674 3,105 0.620 400 1.23 56 190 0.3 240 10

C32008 EA9 1.059 342 0.080 38 0.42 23 336 30 129 302010 JW34 0.983 53 0.055 10 2.26 42 43 7 50 101991 VG 1.027 1 0.049 0.2 1.45 55 25 0.1 74 0.92009 BD 1.063 107 0.052 48 1.27 63 317 10 253 0.062006 RH120 1.033 107 0.024 21 0.60 9 10 600 51 0.4

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1055

distribution of five orbital elements (ignoring the mean anomaly)and (ii) eliminates any possible real clusters.

Our solution was to ‘fuzz’ the orbital elements of each real NEOaround its position in 5-dimensional orbital element space in amanner that maintained the local orbital element phase-spacedensity and thereby preserves both the intrinsic NEO orbital ele-ment distribution and the observational selection effects. For eachNEO (k) in the real population we identified its closest neighbor asthe object with the smallest DSH - Dfuzz. We then generated a new‘fuzzy’ synthetic orbit (n) that has DSH 6 Dfuzz with respect to theoriginal orbit as described below.

If all the difference between the original and new orbit is due toa single orbital element (e.g. Da % an ' ak) then we obtainDa; De; Di; Dx, and DX from:

Dfuzz %#Da$AU#1' e$ #8$

% De!!!!!!!!!!!!!!!!!!!!!!!!!!1& #a=AU$2

q#9$

% 2 sin#Di=2$ #10$% 2e sin#Dx=2$ #11$

%!!!!!!!!!!!!!!!!!!!!d03

2 & d042

q#12$

where

d03 % 2 sin#IDX=2$; #13$d04 % 2e sin#PDX=2$; #14$

and

IDX % arccos cos2 i& sin2 i cos DXh i

#15$

PDX % 2 arcsin cos i sinDX2

secIDX

2

" #: #16$

We then generated a ‘fuzzed’ orbit with #a0; e0; i0;x0;X0$, where theelements x0 were generated randomly within the range(x' Dx; x& Dx). Finally, we calculated the DD between the originaland the fuzzed orbit and repeated the generation of the fuzzed orbitfor the object until DD < Dfuzz. We also repeated the synthetic objectgeneration if the new orbit was not a NEO (had perihelion

a0#1' e0$ > 1:3 AU), was hyperbolic (i.e. e0 P 1), or unphysical (e.g.a0 6 0; e0 6 0). We generated 1000 instances of these ‘fuzzy-realNEO models’ to be used for establishing the statistical significanceof our NEO clusters.

We tested the generation of the fuzzy-real NEO models by gen-erating a series of models fuzzed by D0fuzz % ffuzzDfuzz with ffuzz % 0:0,0.2, 0.5, 1.0, 2.0, and 4.0 and verified that the models behave as ex-pected and as ffuzz ! 0 the generated model reproduces the inputmodel exactly.

The remaining problem with the fuzzy-real NEO models is thatif the real NEO population contains real families then so will thefuzzy-real NEO models. We needed to remove any real NEO fami-lies from the model first—but this is difficult to accomplish whenthere are no known real NEO families. Instead, we used our owncluster results agnostically with the assumption that it does notmatter whether the clusters we identified are real or not, all thatmatters is how often the fuzzing process generates false clusters.Thus, (i) using the cluster detection thresholds determined withthe synthetic NEO population described in Section 2.3.1 we identi-fied all clusters containing P3 members and (ii) treated the largestmember of each cluster as any other NEO as described above but(iii) fuzzed the orbits of the 18 smaller members of the clusterswith ffuzz % 10. We needed to keep the 18 small objects in the mod-el because they represent about 0.3% of the total known NEO pop-ulation—about equal to the 3r contribution to the NEO model thatwe were attempting to measure. On the other hand we needed tokeep them in roughly the correct location in the NEO orbitdistribution.

Fig. 1 shows that the fuzzy-real NEO models’ semi-major axis,eccentricity and inclination distributions match the known NEOpopulation far better than the synthetic NEO model. Even moreimportantly, Figs. 3 and 4 show that the fuzzy-real NEO modelspreserve both the micro and macro DSH distributions (respectively)of the real NEO population that are critical to using the models toestablish the statistical significance of the NEO clusters.

Note that the DSH distribution of the real NEOs is systematicallyslightly higher than the fuzzy-real NEO model at small DSH in Fig. 3.This would be the expected signature if there were real families inthe real NEO population. The data point in the lowest DSH bins inFig. 3 represent two separate pairs of NEOs. If these two pairs werereal it would imply that they were statistically significant but itturned out that they were subsequently identified by the MinorPlanet Center as corresponding to the same physical object. Thus,it is reassuring (1) that our fuzzy-real NEO model DSH distributionagrees well with the real NEOs and (2) that our technique success-fully identified identical NEOs.

3. Results and discussion

The shaded regions in Fig. 5 show the results of running ourcluster identification algorithm on the 1000 family-free fuzzy-realNEO models described above. We searched for clusters containingP4 members over the range of Dcluster–Dpair space with 0:040 6Dcluster 6 0:060, where Dpair 6 Dcluster using fixed SCRmin % 0:75 andPFmin % 0:5 as described in Section 2.3.2. It is surprisingly easy tocreate NEO clusters with P4 members. The shaded areas in the fig-ure correspond to regions where we detected a total of a minimumof 3, 46 or 317 clusters in the 1000 models. The regions correspondto 3-, 2- and 1r confidence levels on the detection of clusters in thereal NEO population.

Fig. 5 also shows the location of our detected C1, C2 and C3 clus-ters in the real NEO population. The clusters can be identified overa limited range of Dcluster–Dpair with the most statistically significantpoint typically having the smallest Dcluster and Dpair . It may be sur-prising that the clusters are not identified in a broad fan extending

Table 2Absolute magnitude (H), diameter (DS and DC ) and DSH for the members of three NEOclusters (C1, C2 and C3) with four or more members. Members of the clusters arelisted in order of decreasing size. None of the clusters are statistically significant at> 3r. DSH and its uncertainty were calculated using the orbital elements andassociated errors from www.jpl.org as shown in Table 1. The diameters DS and DC

were calculated using albedos of pV % 0:2 and pV % 0:03 corresponding to the meanalbedos of S and C class asteroids as reported by Mainzer et al. (2011).

Name H (mag) DS (m) DC (m) DSH DDSH #,10'6$

C12000 HW23 18.5 590 1500 – 42006 JU41 19.3 410 1100 0.041 62001 PF14 19.5 370 1000 0.032 42008 LN16 19.9 311 803 0.048 4

C21999 YD 21.1 180 460 – 802003 UW5 24.3 40 110 0.051 32008 UT5 24.5 40 100 0.044 32008 YW32 25.2 30 70 0.049 102007 YM 26.2 180 40 0.052 16002005 WM3 27.7 8 20 0.026 170

C32008 EA9 27.7 8 22 0.039 42010 JW34 28.1 7 18 0.050 21991 VG 28.5 5 15 – 2702009 BD 28.8 5 13 0.050 105

2006 RH120 29.5 3 9 0.049 105

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upwards and to the right in the figure. The truncation in the regionin which they are identifiable is due to the SCRmin % 0:75 andPFmin % 0:5 cuts—using looser values of Dcluster and Dpair allows thecluster to ‘absorb’ other nearby objects but these objects then typ-ically drive the cluster’s SCR and PF values below the SCRmin andPFmin thresholds.

Fig. 5 shows that the C1 cluster is statistically significant at! 2:0r. The C2 and C3 clusters are likely even less significant basedon their large Dcluster and Dpair error bars extending well into theinsignificant region of the figure. It is important to remember thatour calculated statistical significance of the NEO clusters hinges onthe reliability of our fuzzy-real NEO models described in Section2.4.1.

Given that all four C1 members are relatively large NEOs withH < 20 we repeated the entire search for clusters using only thoseNEOs with H < 20 and re-calculated their statistical significance.As expected, the statistical significance of the C1 cluster increasesto about 3r because of the reduction in the total number of NEOsto ! 3100.

While we have taken every precaution in developing our NEOmodels we understand that they have their limitations. For instance,the statistical significance of the NEO clusters increases/decreases ifwe ‘fuzz’ the model more/less #ffuzz > = < 1$. We assume that the‘natural’ value is ffuzz % 1 and use this value when quoting the statis-tical significance of our NEO clusters. In the remainder of this workwe consider only the C1 cluster as a candidate NEO family.

Fig. 4. DSH distribution for the closest neighboring objects in the real NEO population and the average ± RMS for 1000 fuzzy-real NEO models.

Fig. 3. DSH distribution of close NEO pairs normalized by the total number of possible pairs in each population. The solid black histogram represents the real NEO populationwhile the black data points represent the average ± RMS of 1000 ‘fuzzy’ real NEO models (the ‘fuzzing’ technique is described in Section 2.4.1). The gray histogram representsthe nominal synthetic NEO population while the gray data points represent the average ± RMS of 1,000 ‘fuzzy’ synthetic NEO models.

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1057

3.1. Dismembering the C1 cluster

In this sub-section we perform several tests of the C1 cluster todetermine if it is consistent with being a genetically related NEOfamily.

3.1.1. Provenance and taxonomyDespite our minor caveats with the Bottke et al. (2002) NEO

model described above it can still be used to estimate the probabil-ity that a NEO derives from one of their five NEO source regions.The C1 cluster falls in the NEO model bin with centrala % 2:150 AU, e % 0:425 and i % 7:5 deg for which objects have!35% probability of deriving from the m6 region and a !63% chanceof having evolved from the so called ‘Mars Crossing’ region. Inother words, a nearly 100% probability of deriving from a sourcein the inner region of the main belt.

The inner region of the belt is dominated by S-class asteroids(e.g. Zellner, 1979) so it is a little surprising that SDSS spectropho-tometry (from the 4th Moving Object Catalog, MOC4; Ivezic et al.,2002b) for the only available C1 cluster member, 2000 HW23, is notS-like.5 We performed a simple linear interpolation and extrapola-tion of the 2000 HW23 SDSS MOC4 5-band spectrophotometry anderror bars to the SMASSII bands’ central wavelengths (Phase II ofthe Small Main-Belt Asteroid Spectroscopic Survey; Bus and Binzel,2002). (The only extrapolation was from the real data point at892 nm to the data point at 920 nm.)

Even if at first glance the asteroid appears to be most similar tothe O-type due to the dip in reflectivity at long wavelengths (seeFig. 6) a normalized v2 fit of the 2000 HW23 spectrophotometryto 26 major asteroid classes from A through Xk (from Bus and Bin-zel (2002)) suggests that it is most consistent with the unusual B-type, a subset of the larger C-complex. The O-type provides a closesecond best fit. In general, keeping in mind that the fit used dataextrapolated from the SDSS MOC4 catalogue, 2000 HW23 yieldsgood fits with the different types in the C-complex, not the S-classthat dominates the inner main belt.

The case for the C1 cluster being a legitimate family will bestrengthened if the spectra of other C1 cluster members can beshown to be similar to 2000 HW23 and belonging to the C-complex.

3.1.2. Size–frequency distributionUnder the assumption that C1 is a family we can estimate the

slope of the family’s size–frequency distribution (SFD) assumingthat the SFD is proportional to 10aH . An unreasonable value ofthe slope would indicate that the cluster might not be a geneticfamily. Converting the observed C1 cluster member’s H values intothe true SFD requires the detection efficiency as a function of abso-lute magnitude, !#H$, for objects with C1-like orbits. We estimated!#H$ using the synthetic survey simulation described in Section2.3.1 by dividing the number of derived objects with C1-like orbitsby the number of objects in the input model. The survey simulationdoes a good enough job of reproducing the real NEO population forthe purpose of estimating this efficiency and will not dominate theinduced error on the measured SFD. We found a reasonable fit tothe efficiency from the synthetic data using

!#H$ % !0

1& exp(#H ' L$=w)#17$

with nominal values of !0 % 1:0; L % 20:0 and w % 0:76. A maxi-mum likelihood fit to the four C1 members’ H distribution yieldsa % 0:23&0:10

'0:08 where the error bars are statistical-only. The statisticalerrors on the fit are much larger than the systematic errors intro-duced by the uncertainty on the efficiency function parameters.

The slope of the C1 cluster’s SFD (Fig. 7) is shallower but notinconsistent with the overall NEO population’s SFD of 0:35* 0:02measured by Bottke et al. (2002) for H < 22 and it is surprisinglyclose to the value of 0:26* 0:03 for H J 18 from Mainzer et al.(2011). Parker et al. (2008) measured the SFD for many main beltfamilies and found that they can have complicated SFDs that can-not be fit by a single power-law and that they have a wide range ofslopes. The slope varies from 0.35 to 0.97 for single-slope familiesand from 0.10 to 0.62 at the faint end of the H distribution for fam-ilies with a broken power law SFD (i.e. H values approaching thoseof the C1 cluster members). Furthermore, Richardson et al. (1998)

Fig. 5. Statistical significance of the C1, C2 and C3 clusters as a function of the Dcluster and Dpair thresholds with SCRmin % 0:75 and PFmin % 0:5. The shaded gray regions markareas of 3-, 2-, and 1-r significance for clusters with P 4 members using the fuzzy-real NEO models (see text for details). The star, circle and triangle represent the moststatistically significant location of the C1, C2 and C3 clusters respectively. The small black lines on the C1 data point represent the derived errors in Dcluster and Dpair due to theorbital element errors. The long vertical and horizontal lines through the C2 and C3 cluster’s data points represent the same errors for those clusters. The white region underthe diagonal line is unphysical with Dcluster < Dpair .

5 We used the solar colors from Bilir et al. (2005) and Allen et al. (2005).

1058 E. Schunová et al. / Icarus 220 (2012) 1050–1063

suggests that tidally disrupted asteroid families have shallow SFDs.Thus, we cannot use the SFD slope to exclude the hypothesis thatthe cluster is a family since the measured value is consistent withslopes of established families.

3.1.3. Backward orbit integrations and tidal disruption at MarsIf the C1 cluster is the remnant of a recently formed family then

we expect that a backward integration of the four member’s nom-inal orbits would show a rapid convergence to a single orbit andthe time frame for the convergence would indicate the time ofthe family’s formation (e.g. as observed in the Karin family;

Nesvorny et al., 2006b). On the contrary, Fig. 8 shows that the nom-inal orbits are similar but undergo a gradual dispersal moving100 ky into the past, i.e. they appear to have converged to theirmost similar orbits at the present time. The semi-major axis,eccentricity and inclinations of the members remain similar butthe longitude of perihelion and ascending nodes gradually diverge.The effect is summarized in panel Fig. 8F showing the evolution ofthe cluster’s DSH with time. We note that two pairs of objects re-main tight in their mutual DSH for the entire integration.

Given the apparent stability of the cluster members’ orbits weintegrated all four nominal orbits backwards for 10 myr and found

Fig. 6. Spectrophotometry of the C1 cluster’s largest member, 2000 HW23, from the SDSS MOC4 by Ivezic et al. (2002b) (black points) and the corresponding interpolated andextrapolated data points and error bars at the SMASSII filters’ central wavelengths Bus and Binzel (2002) (gray triangles and connecting lines). The synthetic SMASSII datapoints and lines are offset by '0.02 for clarity. The thin black curves and their adjacent gray curves represent the mean ± RMS respectively of five asteroid taxonomic typesfrom the SMASSII survey. All distributions have been corrected to solar colors and normalized to 1.0 at 550 nm. The relative reflectances of the C, O, B, S and V types are offsetby +1.5, +1.0, +0.5, '1.0 and '1.5 respectively.

Fig. 7. Cumulative absolute magnitude distribution of the four C1 cluster members (gray). The thick black curve represents the product of the efficiency function (see text)with the result of the maximum likelihood fit to the SFD with a % 0:23&0:10

'0:08.

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1059

that they are exceptionally similar back to 1.5 Myr in the past. Themutual orbital stability is unusual for NEOs given that the numer-ical integrations showed numerous close encounters of all objectswith Mars during this time. The lack of convergence of the nominalC1 members’ orbital elements suggests that they are not geneti-cally linked. To firmly establish the statistical significance of theC1 cluster’s genetic relationship would require the generation ofhundreds of clones of each C1 member and similarly generatingclones for a large number of C1-like clusters.

Rather than explore that computationally challenging route weexplored a few instances of the opposite question—is it possible thata family producing event in the recent past can reproduce the ob-served C1 cluster’s orbit and DSH distribution? We created 500 clonesfor each of the four C1 members where each clone is consistent withthe available astrometry (the clones were generated using the covari-ance sampling technique in the OpenOrb orbit-computation package;

Granvik et al., 2009). We integrated all the clones backwards in timefor 100 ky and discovered that all of the objects suffered several closeapproaches with Mars and a large number of clones approached towithin Mars’ Hill sphere. Since the C1 cluster is composed of Amortype NEOs we did not expect nor identify any close approaches tothe Earth and Venus.

One 2000 HW23 clone passed inside Mars’ Roche limit!70,800 years ago and is therefore a candidate for tidal disruptions(assuming that the parent body is a spherical fluid or rubble pilewith a density q = 1.95 g cm'3 like (25143) Itokawa (Abe et al.,2006), Mars’s Roche limit is !3.08 Mars radii). This result needsto be tempered with a comparison to the likelihood that any NEOwill approach to within Mars’ Roche limit in the same time frame.We found that 5070 of 8133 known NEO’s nominal orbits approachMars to within 1 Hill radii at least once during the last100,000 years. Assuming that close approaches to within the Roche

Fig. 8. Panels A–E: Backward evolution of the C1 cluster members’ orbital elements from the present (t0) to 100 ky in the past. Panel F: Dcluster evolution for the C1 cluster fromt0 to 100 ky in the past. Dcluster was calculated with respect to 2000 HW23 and therefore it does not appear on the panel. Dcluster with respect to 2000 HW23 does not differsignificantly from the value calculated with respect to the average orbit of the C1 members.

1060 E. Schunová et al. / Icarus 220 (2012) 1050–1063

limit are as likely for clones of all NEOs (1 in 2000 C1 clones) thenwe might expect that about 2.5 of the known NEOs have actuallyapproached to within Mars’ Roche limit and could be candidates

for tidal disruption. Perhaps the C1 cluster’s hypothetical parentbody is one of them and we will represent it as 2000HW 23—theclone that made the closest approach to Mars mentioned above.

A

B

C

D

E

F

Fig. 9. Panels A–F: Orbital elements and Dcluster distribution for the C1 cluster members (dark gray) and two sets of six secondary 2000HW 23 clones (see text for description).The two sets of clones have D~vx , or D~vy , or D~vz with respect to 2000HW 23 of ±1 m s'1 (light gray) and ±5 m s'1 (white) respectively at the time of closest Mars approach. Thereare only three values for the real C1 cluster because Dcluster was calculated with respect to 2000 HW23. The Dcluster values for both sets of six secondary clones were calculatedwith respect to 2000HW 23 that was integrated simultaneously.

E. Schunová et al. / Icarus 220 (2012) 1050–1063 1061

To study the properties of a family of objects created by tidaldisruption by Mars we created 12 2000HW 23 clones at the momentof its closest approach of 0.63 Mars Roche radii. The 12 secondaryclones have D~vx, or D~vy, or D~vz of ±1 m s'1 and ±5 m s'1 with re-spect to 2000HW 23. Simulations suggest that the tidal disruptionprocess creates fragments with relative speeds of <1 m s'1 (KevinWalsh, personal communication) so that our integrations willoverestimate the spread in the orbital elements of tidally disruptedfamilies. All 13 objects (2000HW 23 and its 12 secondary clones)were integrated forward from the time of closest encounter tothe present epoch with the same Bulirsch–Stoer integration rou-tine and same time step as our other integrations.

Panels A–E in Fig. 9 show that the values and ranges of a; e and ifor the ±1 m s'1 and ±5 m s'1 groups of secondary clones are com-parable to the values and ranges of the real C1 members. Theranges of X and x, i.e. the spread of values within the two setsof 2000HW 23 clones and the four C1 cluster members, are also sim-ilar. The values of X and x for the 2000HW 23 clones and the four C1cluster members do not need to agree at the present time becausethe 2000HW 23 secondary clones were created from a single 2000HW23 clone whose orbit is slightly different from 2000 HW23 dueto the former’s deep approach to Mars (their semi-major axes dif-fer by 0.05 AU). The difference between X and x for the 2000HW 23

clones and the four C1 cluster members is thus due to their differ-ent precession rates that are sensitive to the initial values of semi-major axes (Murray and Dermott, 1999). Note that the actual C1cluster member’s values show evidence of slightly more spreadingthan the 2000HW 23 secondary clones. Indeed, Fig. 9F shows thatthe distribution of Dcluster for the 2000HW 23 secondary clones is alsotighter than the C1 members. Both observations can be explainedas being due to non-gravitational forces acting on the real objectsover the course of the last !71 ky.

The distributions of orbital elements and Dcluster in Fig. 9 sug-gests that the C1 cluster is consistent with an origin in the tidal dis-ruption of an asteroid during a tidal disruption event with Mars70,800 years ago. If true, the C1 cluster is a very young family. Ina followup paper to this one we will show that the lifetime of tid-ally disrupted families created by Mars are as long as !1 Myr so ifthey exist it may not be surprising that we are beginning to detectthem. Furthermore, if there is a tidally disrupted asteroid family inthe NEO population it seems more likely to be a C-class asteroidand, ceteris paribus, will disrupt more easily since they have a lowerbulk density than the S-class. (Richardson et al., 1998). Thus, it isinteresting that one C1 member with spectrophotometry, 2000HW23, may be in the C-class (see Section 3.1.1).

3.1.4. Proper element cluster searchWe also performed a cluster search with NEO proper elements

(Gronchi and Milani, 2001). Since the C1 cluster was detected asa cluster in osculating element space it seems plausible that itshould also be detectable in proper element space as describedin the introduction. A complication arises because our clustersearch algorithm uses a 5-element DSH-criterion that incorporatesthe longitude of the ascending nodes and argument of perihelion(the d3 and d4 terms in Eq. (1)) while only the proper semi-majoraxis, eccentricity and inclination #ap; ep; ip$ can be calculated forasteroids. Thus, the DSH for clusters in proper element space, Dp,must be smaller than those found using the 5-element DSH withthe osculating elements—Dcluster % 0:002 compared to Dcluster %0:040 used for the osculating element cluster search.

The proper element cluster search revealed that the C1 cluster isnot outstanding among NEO clusters. We identified a C1p cluster inthe proper element search that has three objects in common withthe C1 cluster—2000 HW23, 2001 PF14, and 2008 LN16—and thereare 166 clusters with 5 or more members that are bound tighterthan C1p with Dp % 0:018. The C1p cluster contains two additional

objects, 2002 RA182 and 2004 TE18, instead of 2006 JU41. The oscu-lating element DSH for 2004 TE18 and 2002 RA182 calculated withrespect to 2000 HW23 are 0.471 and 0.125, much too large to indi-cate an orbital similarity.

We found that the stability of the NEOs in the C1 and C1p clus-ters is not unusual. We computed the MOID with all the planetsfrom Mercury to Neptune during the secular evolution of 8650NEOs (NEODyS, 29/2/2012) according to the dynamical model usedin the computation of the proper elements (Gronchi and Milani,2001). Roughly 5% of all NEOs (429) do not cross the trajectory ofany planet and ! 42% of them (3636) cross only Mars’ trajectoryduring their secular evolution.

The six asteroids in the C1 and C1p clusters are of the latter type.Table 3 shows that all the cluster’s members remain far from boththe Earth and Jupiter, the major NEO perturbers, during their entiresecular cycles. Of the total 3,636 Mars crossing objects we find that

" ! 50% of them have min#MOIDEarth$P 0:1 AU, andmin#MOIDJupiter$P 1 AU," ! 16% of them have min#MOIDEarth$P 0:2 AU, and

min#MOIDJupiter$P 2 AU.

Thus, even if some of them are affected by mean motion reso-nances it appears that many Mars crossing NEOs remain far fromclose encounters with, and are not affected by perturbations from,the Earth and Jupiter. A close approach with Mars must be verydeep to cause a significant perturbation.

Thus, there are islands of long-term stability in the NEO orbitalelement phase space that might ‘collect’ NEOs in a manner thatcould mimic the appearance of a NEO family. The Bottke et al.(2002) NEO model cannot accurately model these regions of stabil-ity because of its low resolution in the orbital element phase space,the relatively small number of objects that were originally used tomap out the NEOs’ residence time probability distribution, and be-cause it is restricted to just three of the six orbital elements.

4. Conclusions

We searched for genetic asteroid families in the NEO populationusing the method proposed by Fu et al. (2005) based on identifyingclusters of objects with similar orbits. We enhanced the method’sutility by developing a technique for assessing the statistical signif-icance of the identified clusters using 1000 realistic family-freefuzzy-real NEO orbital element models. We created our NEO mod-els by cloning members of the known NEO population in a naturalway based on the ‘distance’ between each member and its nearestneighbor. The technique identified three clusters of four or moreNEOs among the orbits in the mpcorb.dat data base. None of theclusters are statistically significant at P 3r and we conclude thatthere are as yet no identified families in the NEO population.

The most statistically significant cluster, C1, contains four ob-jects all with H < 20 and well-determined orbits with the largestmember being asteroid 2000 HW23. We performed several addi-tional tests of the C1 cluster’s family veracity including checking

Table 3Minimum MOID with the Earth and Jupiter for the NEOs in the C1 and C1p clustersduring the course of their secular evolution.

Name min#MOIDEarth$ #AU$ min#MOIDJupiter$ #AU$

2000 HW23 0.26782 2.161122001 PF14 0.27176 2.232252006 JU41 0.23183 2.187232008 LN16 0.26740 2.188862002 RA182 0.27768 2.162642004 TE18 0.26278 2.23812

1062 E. Schunová et al. / Icarus 220 (2012) 1050–1063

the members’ taxonomic identification, their bias corrected size–frequency distribution, the possibility that they originated in afamily-producing tidal disruption at Mars about 71 ky in the past,and whether it is also identifiable as a cluster in proper elementspace. None of these tests exclude the possibility that C1 is a genet-ically related family but at the same time none of the tests providesufficient evidence to elevate the cluster to family status.

The search for families amongst the NEO population is clearlynot as straightforward as the same search amongst the main beltpopulation. Special care must be taken when assessing the statisti-cal significance of a purported NEO family especially with regard toaccounting for observational selection effects in the NEO popula-tion. At the very least, mere similarity of the orbital elements asevidenced by the DSH criterion being less than an arbitrary andcommonly used value like 0.2 is insufficient for deciding uponany genetic relationship between NEOs and, by extrapolation,between NEOs and meteors.

Acknowledgments

This work was supported by NASA NEOO Grant NNXO8AR22G.M.G. was also funded by Grants #136132 and #137853 from theAcademy of Finland. We acknowledge CSC-IT Center for ScienceLtd. for the allocation of computational resources. Schunova’s workwas also funded by The National Scholarship Programme of SlovakRepublic for the Support of Mobility of Students, PhD Students,University Teachers and Researchers and VEGA Grant No. 1/0636/09 from the Ministry of Education Of Slovak Republic. We alsothank several colleagues who provided insight into NEO family is-sues including Kevin Walsh, Giovanni Valsecchi, Alessandro Mor-bidelli, Patrick Michel, Joseph Masiero, Davide Farnocchia, andSeth A. Jacobson.

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